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Difficult problem involving a percentage of the period of a sinusoid

Im having difficulty intuitively understanding how to solve this problem: $x(t) = A*cos(\Omega*t + \phi)$ $A > 0$ $\phi$ range is $(−\pi,\pi]$. $x(t) ≥ 2.4$ for $18$% of each period takes ...
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7 views

Implementing Normalized Cross-Correlation using FFT - How to?

Is there any way to calculate the normalized cross correlation between 2 signals by using the FFT? (I managed to implement it already for standard cross correlation equation). Thanks in advance,
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14 views

Is constant system a Causal System?

Is y(t) = 1 a causal system? From the definition of causal systems , a causal system is a system where the output depends on past and current inputs. Here the system doesn't depend on any input. ...
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2answers
17 views

Is this system Causal?

The output system is: $x(t)$ -->(S)--> $y(t) = \int_{-\infty}^{t}x(\tau) d\tau$ Recall that the system is causal if the output at $t$ depends only on input before $t$, or if the impulse response ...
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1answer
42 views

building time signal after inverse FFT

I have managed to implement both forward and inverse FFT transforms in C#. And, i tested them by taking signal both ways and got the real part after iFFT equals the original signal. Now , I have ...
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20 views

non negative matrix factorisation [closed]

i am working on a project involving the use of non negative matrix (NMF) for the separation of a mixture of audio signals.can please give me the mathematical explanation of NMF so that i can ...
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15 views

Mathematical Expression for a Fourier Transform $s(T)$

$S(f)$ is the Fourier transform of a non-periodic signal, $s(t)$. $S(f)$ is given by: $S(f) = 1,$ for $−1/2 ≤ f ≤ 1/2$ and $0$ otherwise. What would be a mathematical expression for $s(t)$?
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1answer
365 views

Wavelet or FFT for Transient signal analysis?

For now I use FFT to analyze the response of an electrical system to some transient signal. The transient signal is $x(t)$, which translates to $X(w)$ in the frenquency domain. On the other hand I ...
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18 views

When studying 2D gabor functions why is a gaussian called elliptical?

Consider $$G(x,y)=\frac{1}{2\pi\sigma\beta}e^{-\pi\left[\frac{(x-x_0)^2}{\sigma^2}+\frac{(y-y_0)^2}{\beta^2}\right]}e^{i[\xi_0x+\nu_0y]}.$$ This is the product of a complex plane wave and what this ...
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1answer
19 views

Specific question about downsampling in frequency domain

I'm confused about why Equation 4.74 can be expressed as Equation 4.76 after the summation index is expressed as in Equation 4.75. Could somebody please explain? Thanks! Downsampling DTFT
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1answer
686 views

Plotting discrete time signals involving sumations in matlab.

Many of the examples I've encountered while looking for an answer are simple functions that do not involve summations. Suppose I have the following function; ...
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1answer
24 views

Is interpolating well-sampled data (Nyquist-Shannon theorem) a cheat?

Suppose to sample a signal $s(t)$ with bandwidth $B$ with a sampling frequency $f_c$. Suppose also that the number of sample collected is $N$ (the duration of the signal acquisition is then $T = ...
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29 views

What is the requirement for separable parameters in an LSQ fit?

I am trying to determine the amplitude of a sinus modulated sinus as accurate as possible. My sampling frequency is sufficently high. The entire model looks as follows: $$ ...
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1answer
22 views

Why is autocorrelation used without normalization in signal processing field?

According to the wikipedia(Link), autocorrelation has two definition. Oh my god! In statistics, the definition of the autocorrelation between times $s$ and $t$ is like the following: $$\displaystyle ...
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2answers
114 views

How can I recover a sequence of numbers given a corrupted version of it?

I have an unknown sequence of real numbers $x_i$ and a known sequence of real numbers $y_i$; $y_i$ is a corrupted version of $x_i$, i.e., $$y_i=x_i+n_i$$ where $n_i$ is a random number distributed ...
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1answer
45 views

Sampling the Sine Function

Consider the sampled sine function, $f(n)=\sin(\omega n)$, where $n$ is an integer. If $\omega_2 = 3\pi/2$, does there exist an $0 \leq \omega_1 \leq \pi$ such that $\sin(\omega_1 ...
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1answer
31 views

Orthogonality of Two Signals.

My last question's link: Reconciling different definitions of orthogonality However, I failed to understand why they are equivalent. If $f$ and $g$ are real, \begin{align} ...
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1answer
46 views

Reconciling different definitions of orthogonality

I want to establish about orthogonality in my mind. I knew the orthogonality of two functions $f$ and $g$ in interval T like the following: $$ \int_{<T>}f(t)g^*(t)~dt=0 \tag{1} $$ where $$ ...
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1answer
31 views

Application of Residue Theorem to inverse Fourier transform

I'm reading through a derivation in a book and am having trouble understanding a step. Here's a screenshot 3.46 is the equation in $(k,\omega)$ space. They're doing an inverse Fourier transform back ...
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1answer
35 views

Fast Fourier Transform as Matrix Factorization

I'm given a vector of length 4 and three matrices that correspond to a Fast Fourier Transform, I'm not exactly sure which one, but I guess it's supposed to be the Cooley-Tukey algorithm. Here is the ...
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1answer
10 views

Unit of the second derivative of the power spectral density

To characterize a subtle oscillation embedded in a time varying voltage signal measured in microvolts, I took the second derivative of the PSD (which I computed as the fft of the autocorrelation) ...
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0answers
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How to create obtain aliased version of $f(t)$ by upsampling whenever $f(t)$ at every $t$ is available

Suppose there is original complex-valued $f(t)$ with $t$ ranging from $-\infty$ to $\infty$. It is possible obtain samples from original $f(t)$ at every $t$ with some negligible error. If one ...
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1answer
45 views

Why do high-frequency dynamics quickly go away in a step response?

As we know, a step input hits all the frequencies of a dynamical system. However, my professor told me today that the high-frequency response is only present for a short time at the very start, and ...
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0answers
69 views

Sampling theorem - signal reconstruction

Hi and thank you for your help. I got stuck for the last days with this chapter in this book https://ccrma.stanford.edu/~jos/mdft/Sampling_Theorem.html. What I do not understand is ...
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27 views

Eigenvalues for correlation matrix which have the form of an harmonic function

As a continuation to this question, I took the matrix $C_{2 \times 2}$ which is: $$C=\left[ \begin{array}{} a& ace^{-\frac{|\phi_1-\phi_2|}{2}}\\ ace^{-\frac{|\phi_1-\phi_2|}{2}} ...
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1answer
430 views

Kalman Filter Process Noise Covariance

I want to model the movement of a car on a straight 300m road in order to apply Kalman filter on some noisy discrete data and get an estimate of the position of the car. In a Kalman filter the matrix ...
2
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1answer
814 views

How do I compute “AUC” Area under the curve number, if all I have are my TPR and FPR values?

I am trying to rank my neural network, which is trained for binary classification. That is, given a set of input signals, it outputs either a 1 or a 0. I have a training set, where I have the actual ...
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0answers
19 views

Is it possible to use regularization to minimize the (expected) number of non-zero digits in a number?

This question may be slightly related to this question on length of the representation of a number in a certain basis. Introduction / Background In image and video coding, particularly the ...
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0answers
7 views

How do I validate my ARMAX model?

Say I have some ouput $y_1, y_2, \ldots, y_N$ and inputs $x_1, x_2, \ldots, x_N$ which, by various time series methods, I've found to match an ARMAX(2,2,1) model. So I've found the estimations for ...
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0answers
19 views

Variance of $\hat{b} =\underset{b}{\mathrm{argmin}} \sum_{t=1}^N [y_t - bu_t]^2 $

We have $y_t = bu_t + e_t$ where $u_t$ is the input signal and I'm trying to find an expression for the variance of the estimate for b that is determined to be $$\hat{b} = ...
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0answers
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Resampling a digital sound signal [migrated]

As a non maths-pro, I'm looking for some pointers. I am rewriting the audio core in my emulator to improve accuracy, and am getting a bit stuck on the specifics of the technique. I am trying to ...
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36 views

What is the output $y(t)$ when you have input $x(t) = \cos(2 \pi t) $ and frequency response response $h(t) = u(t) - u(t - 1/2)$?

The output $y(t)$ is the convolution of input $x(t)$ with impulse response $h(t)$: $$ y(t) = h(t) * x(t) $$ This is a linear, time invariant system. What is the output $y(t)$ in real form when you ...
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2answers
2k views

Undecimated Wavelet Transform (a trous algorithm) - how to determine 'anchor'/'center' of convolution filter

i am currently implementing the 'Undecimated Wavelet Transform' with the 'a trous' algorithm. See e.g. http://www.znu.ac.ir/data/members/fazli_saeid/DIP/Paper/ISSUE2/04060954_2.pdf, section II-A. As ...
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0answers
46 views

Solving a non-linear parametric equation

I am interested in solving a parametric equation where the unknown function is a function of time, and there is also an input. For example: $ y^{2}(t) + y(t) = \sin(t)$ I am coming from a signal ...
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3answers
387 views

What is a moving average system?

Can someone elaborate on what a moving average system is? I know that the system is defined as: $$y[n] = \frac{x[n] + x[n-1] + x[n-2]}{3}$$ How would we draw $y[n]$ given that we have a graph with ...
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1answer
8k views

Hamming Window - Understanding Formula

I'm trying to understand the Hamming window concept (Signal Processing) and the equation looks like the following: Where: N = number of Samples in each frame Y[n] = Output Signal X(n) = Input ...
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2answers
758 views

Matlab: Impulse response of linear time invariable (LTI) sine-signal

I'm preparing for a lab in a Signals and Systems course in my university, 5th semester. I've found old exercise material from the class and since I know some Matlab and have dealt with LTI systems ...
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2answers
35 views

Do decaying exponential signals have finite energy?

"A signal that decays exponentially has finite energy, so, it is also an energy signal." http://www.songho.ca/dsp/signal/signals.html I don't quite get how that can be true. Energy of a ...
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0answers
36 views

Why does inputting complex exponentials into a system give its frequency response?

Let's say I have an FIR filter with the equation: $$ y[n] = \sum_{i=0}^{N-1} h[i] x[n-i] $$ I know that to find the frequency response of this filter, I need to input a complex exponential in place ...
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0answers
19 views

Single Sideband LSB-SC Demodulation

The problem is how the phase φ effects the outcome when the input(message signal) is the DSB-SC LSB. It's : message: $m(t)=A_{m}cos(ω_{m}t)$ carrier: $c(t)=A_{c}cos(w_{c}t)$ I found that the LSB ...
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0answers
31 views

Can downsampling create energy at the Nyquist frequency?

I am a bit surprised by the following and would like to share it with you. I expect I am mistaken somewhere and will be happy to be corrected. I have searched StackExchange not only in Mathematics ...
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0answers
21 views

How to find out the Power of $x(t)$?

I am studying signals and system. I learned that \begin{align} P&=\lim_{L\to\infty} \frac 1{2L} \int_{-L}^{L} |x(t)|^2 dt\\ P&=\frac 1{T} \int_{<T>} |x(t)|^2 dt ~~~\mbox{, P could be ...
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1answer
17 views

In signals processing why is the discrete sequence x[n] undefined (as opposed to 0) when n is not an integer?

In Oppenheim & Schafer's "Discrete Time Signals Processing" it's written that: ... it is important to recognize that x[n] is defined only for integer values of n. It is not correct to think of ...
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28 views

Why is this defined as $u[n]$?

In LTI systems there are two famous functions the unit step function and the unit impulse functions. And they are defined as follows. $$ u[n] = \begin{cases} 1, & \text{if $n$ $\ge$ 0} \\ 0, ...
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1answer
28 views

is there any good way to figure out number of fourier series frequencies of some signal?

Suppose you have $f(t)$, but you do not know the exact function and can only measure $f(t)$ at certain time. Assume $f(t)$ is complex-valued with $t$ being "time." One wishes to find out the number ...
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0answers
18 views

Is it possible to do effectively irrational-interval sampling of a continuous signal?

Suppose there is a real-valued $f(t)$, with $t$ being time. And one wishes to sample at interval of $\pi$, for example. Perfect irrational-interval sampling is not possible, but is there a way to do ...
2
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1answer
5k views

Plotting a one-sided amplitude spectrum

I have a continuous signal $x(t)$ such that $$x(t)=12\cos(6\pi t)+6\cos(24\pi t)+3\cos(30 \pi t)$$ and is asked to sketch a $1$-sided Amplitude Spectrum of the signal $x(t)$ if sampled above the ...
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1answer
14 views

Relationship between short-time and large-frequency asymptotics in Fourier transform

I am trying to understand how the short-time behaviour of a function $f(t)$ influences the large-frequency asymptotics of its Fourier transform $g(\omega)=\mathcal{F}[f(t)](\omega)\equiv ...
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39 views

Signal processing and algebraic geometry

Signal processing is a pretty huge branch of what I would (maybe wrongly) call electrical engineering. I have heard here and there whispers of interesting connections between signal processing - in ...
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19 views

How to sketch frequency response obtained from H(z)?

How to sketch frequency response obtained from H(z)? I'm adding an example and its solution below. I did not understand some of the things for question 7.4, part D. Any help is appreciated! There ...